Chapter_50 - 5 Generating Dependent Random Variables 5.1...

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5 Generating Dependent Random Variables 5.1 Order Statistics X 1 , . . . , X n order -→ X (1) bB±² MIN < . . . < X ( n ) b ² MAX b ² Order Statistics INDEPENDENT SAMPLE order DEPENDENT . We often work with order statistics. Why is this a problem? Just generate X 1 , . . . , X n and then order them! Want to avoid expensive sorting algorithms. How about X ( i ) = F - 1 ( U ( i ) ) where U (1) < . . . < U ( n ) are ordered U ’s This still involves ordering the U ’s, however, there are “tricks” to accomplish this. (a) Generate U 1 , . . . , U n independent U (0 , 1)’s. DeFne U ( n ) = U 1 n n . . . U ( k ) = U ( k +1) × ( U k ) 1 k k = n - 1 , n - 2 , . . . , 1 . . . Then U (1) , . . . , U ( n ) are ordered U (0 , 1)’s. Proof Consider max { n iid U (0 , 1)’s } . cdf of max at y = P( U ( n ) y ) = P( U 1 , U 2 , . . . , U n y ) = n p i =1 P( U i y ) from independence = y n 61
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If we generate U ( n ) by inversion , then U ( n ) = n th root of a U (0 , 1). We’ve proved that U ( n ) = U 1 n n is a r.v. having distribution max { U 1 , . . . , U n } . Now think of 0 1 U ( n ) × The other n - 1 correspond to the order statistics of a sample of size n - 1 uniform over (0 , U ( n ) ). Hence result by recursion.
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Chapter_50 - 5 Generating Dependent Random Variables 5.1...

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